Generic Predictions & New Developments in Loop Quantum Gravity

0:00 I think she will be in Europe in September or October. Actually, it's late in the summer. I think she's going to this conference in the winter. I'm going to the early conference. She can't really respond to it. She's accountable. She does email her. that seems to be a characteristic of people who work in quantum gravity dare I say it not for us quantum emails oh no you do, you're the great exception Roger's the worst but he holds the old time record yeah i don't think he's there yeah he doesn't respond if he doesn't respond to you he doesn't respond to anybody else so you're in good but that's okay by the way are you booked up for this evening I think so I was just wondering if you'd be my guest for dinner at that little restaurant the one we ok if you do that if not then it would be very nice if you could because there's a couple of things I don't unfortunately at the moment because I left it in Fougere like an idiot but it's ok we can probably arrange something if not maybe just coffee or lunch tomorrow there were just a couple of things I wanted to ask you about if possible not quantum gravity related just more plans for meetings and things

2:30 whatever's best for you You've probably heard from Andre that Mark was speaking yesterday afternoon for a rather nice little seminar on applications of fibre bundles. Yeah, which caught the second half off. I mean, well, it would all have been very, very elementary indeed to you, lot. We've heard a lot about moduli, spaces and other elements. What people are... well, Markov wasn't embarrassed and stuff like that. Very interesting, isn't it? Oh, yeah, sure. And also I have a success. So you didn't come to this... I'm really sorry. What happened, I went... I did want to come, but the restaurant where open until 7.15, so it was too late. By the time I finished there, it was nearly 10 to 9. Which is about when I came along. Because you said 8, so I thought... No, no, but of course... The thing that we didn't know was that there was an opening up. But anyway, it wouldn't take me to have an hour to get there. Yeah, I'm sorry. I would like to come. He sounds like an interesting guy. Let's just make a note of the name and try and get some of his records. He has a website, I guess. He has a website, I guess. I'd like to listen to his stuff. It was actually wonderful. It was very packed. The place was very, very, very packed. Is it a very modern cafe? I think it may have been the place I've been before. I think Simon Calypso took me there. It was one of these places where you'd go to your bathroom. There's no place to sit down. Yeah, there were lots of poofs on the ground floor. There were lots of cushions. Oh, it wasn't the place, it wasn't the same place I was thinking of in that place. We were then out very late, so... Well, you did most impressively to be up and on your feet speaking again at 9.30 the next morning.

5:00 I felt... I felt... The scary thing was there was one of these things where you're on stage, Did you have a really big audience? I hope so. It wasn't for me. It was for the... I don't know how many... It was a big public inauguration. That's great. Okay, so it's a pleasure to introduce Lee Smolin. He's coming from Perimeter Institute, which is near Toronto in Canada. Probably you know this institute which is very involved in fundamental physics. And Lee is very interested in many things, including the foundations of physics, and he has special interest about the debate on, for instance, background independence or dependence of theory, which is the modern version of the debate between realism and relationism for space and time. And this is related to his main subject of research, which is quantum gravity. And now he will present some recent results in the field of quantum gravity, which is another way than a string theory to try to find a new fundamental theory and this of course applies to cosmology and we will present recent results of application of this theory to cosmology and we will show that there are some links which could be in some sense expected maybe with quantum information. Thank you for coming. It's a great pleasure to be here, speaking in Paris. I gave a couple of other talks here, and this is a scientific talk, so there's no element of philosophy or anything else here.

7:30 This is quantum gravity and recent results in quantum gravity. The point of view that I want to take in this talk is the following. How do we decide when a quantum theory of gravity is right or at least promising? And of course the answer is experiment. For a long time we've understood a list of criteria that a quantum theory of gravity has to get right in order to be correct. It has to, in an appropriate approximation, coarse-grained approximation, reproduce Einstein equations, for example. But if you look at the history of attempts at unification, it's very clear that theories triumph not because of the things that they're expected to do that they get right, but because of the surprises, the things that weren't expected, in, but that they get right anyway. And so what I want to talk about here is such surprises coming from a class of approaches to quantum gravity which are related to loop quantum gravity. And I say related to, I'll make it clear what I mean by that, but there's some confusion over the use of the term loop quantum gravity and I'll make it clear that I'm talking about actually a large class of theories that I'll find. So the plan of the talk is, I'll talk very quickly about the basic assumptions and by now classic, if you will, results about loop-clinal gravity and related theories. I apologize. For some people this will be an introduction, but I'm not going to give an introduction. I'm just going to define the class of theories and state the main results. My excuse is that this is by now a well-developed subject and I can't go back to the beginning and justify everything, but I will define everything that language. I'm happy to discuss that a little more. Then I'm going to talk about a new point

10:00 of view about the complex of how one gets physics from this class of theories that, as Mark said, tied to quantum information theory. And here I have to talk about works for Team Marco Pulo and collaborators, which is a new method which has been, as you'll see, very fruitful. Then I'm going to talk about two applications coming from that, one of them having to do with particle physics. And this is one of the surprises, and indeed for many of us in the field, it was a surprise. Most of us, all of us in this field have taken the point of view that we're studying the power of quantum gravity and issues of unification with matter, like for example comes through string theory, will come later, will come from some extension of the theory, putting the appropriate degrees of freedom in, making unification with string theory, or something like that. And it turns out all of that was wrong, and what Marco Pujo and her collaborators found is that there already is matter in these theories, leading to a question, can you get the right matter, that is the real world, that we see experimentally? And it turns out that using results of Sundance, Wilson, Thompson, using really a kind of discovery of Sundance, Wilson, Thompson, something remarkable and surprising comes out, which I'll leave for you to judge whether it's coincidence or to be taken seriously. Then I want to talk about very recent work. This was published earlier this year. Very recent work about cosmological implications, also a complete surprise. And I hope that I get at least to state the main ideas and the main results so far. This is work in progress, also with Focini Marco Pulo, a student from Santa Cruz, Chanda Pressback-Weinstein. And this is work in progress, and this is the first time it's presented anywhere. and I hope at least to get to state it, because I'm very excited about it. With this time, and probably there won't be, I'll at least mention doubly special relativity,

12:30 which is also an unexpected, surprising consequence. And I want to mention that all three of these have possible experimental implications, which is why they're worth presenting. Okay, so ludicrous gravity is now a large community, and that was just to show a subset of the names. So it's not just Carlo Rovelli and Abba Ashtucar and myself sitting in the cafe somewhere fantasizing. Okay, so here is the class of theories that I'm going to be working with in all these different applications, and to give them a general name, we call them causal spinel And I'm not going to justify this, I'm just going to define a set of quantum theories and then state some results. So, the input for these theories are, first, pick an algebra. A Lie algebra, SU2, a quantum group could be, say, the quantum extension of SU2 at the root of unity. A super-algebra could be a super-algebra. And then pick a differentiable manifold only up to differential structure, so there are no classical fields in this formulation whatsoever, no metric, no connection, just pick a manifold up to differential structure. This manifold represents the differential structure and the topology of space. Now, we define a Hilbert space by giving a normalizable basis, and the basis elements the following. First of all, pick any graph and consider for that graph possible labeling. The labeling are defined as following. Given the algebra, G, for every edge you label with a irreducible representation of G. For example, if G is SU2, then that's thin. And for every node, you consider the product of the the edges incident on that node and pick an invariant, or what the mathematicians call an intertwiner, and label that with a node. A graph so labeled is called a spin network because the first realizations of this were done by Roger Penrose in the early 60's

15:00 and he used SU2, so labeled with spins. Now, consider every embedding of every such labeled graph in the manifold up to diffeomorphisms. And that's for every graph and each one of those, each diffeomorphism class in the manifold of embeddings of labeled graphs is a normalizable element of the basis. That is the orthonormal basis. And that is the Hilbert space of the theory we're working with. Now in the next slide or two I'll state some results that explain what that possibly has to do with quantum gravity, but those are the Hilbert spaces that define the set of approaches called loop quantum gravity. That's the kinematics, the quantum kinematics. The quantum dynamics can be formulated in a Hamiltonian framework or in a path integral framework. I'm going to talk here about the path integral framework or spin-form framework, which is the one most studied in the last 10 or so years, and here's how it goes. To define a history consists of a series of local moves acting on the graph. So you start with an initial state in the Hilbert space, say a basis state like this, And then the theory tells you that dynamics proceeds by a series of local moves applied to graphs. For example, you can take a three-valent node to a triangle, or you can switch to adjacent, switch the connectivity to adjacent nodes. If they're four-valent, then here are the local moves, a node to a tetrahedra or two adjacent nodes to a triangle. And to define the dynamics of the theory, give us a small set of local moves and assign to each one an amplitude, which is a function in general of the labelings. And then a history, from an initial state to final state, is a sequence of local moves that take the initial graph to the final graph. and the amplitude of the history is the product of the amplitude of the history and then you do a fine and sum over histories to define the climate theory.

17:30 And each history by itself has a causal structure because the moves have a partial ordering. That is, if a move acts on a region which has been created by its previous move then we say that the second is to the causal future of the first there can also be moves that are independent, and so there's a partial order among the moves in each history, and they give a causal structure, which is the causal structure of each common history. Okay, any questions? That is the phononics and dynamics of itself. Question, technically, embedding meaning like monomorphism. Embedding means up to 50 morphisms. And embedding itself like monomorphism, as you just think, just really, you don't see all morphism, from rough to manifold, just monomorphism, right? What do you mean by evading? Yes, I think so. And history is a generally temporal notion? Please, is it generally temporal notion? The temporal notion comes from the partial order among the groups, which I'll be discussing more. The partial ordering, is it a well-ordered? No, it's just a partial order. Because if I have, for example, I can take this node to a triangle and this node to a triangle and they don't affect each other. I could do it in each order. But if I take this node, sorry I don't have a chalk, but if I take this node, say, to a triangle, and I take a node in that triangle and this node and exchange them using, say, this move, then there was a necessary order there. So there's a partial order, not a complete order. Now, what this has to do with the quantization of classical theories of gravity is based on the following theorem, which is, both there is an existence theorem, there is a uniqueness theorem, these theorems, and I'm sorry I forgot to put the references here, are 2004, so it took a long time to get these rigorous theorems, but they're fundamental. So now consider in the same setting a gauge theory, which is dipheomorphism. That means that given the same algebra G, I have the spaces of a gauge connection on the spatial manifold, and this is going to apply when the spatial manifold has dimensions of 2 or greater.

20:00 There's no other field, there's no metric field, there's the connection field, and there can be a conjugate electric field. So we have a connection value in G which is a connection and a conjugate electric field which is a vector density also value in G. There's then an algebra which I didn't write down here so maybe I just I think I have a blackboard here. The algebra is based on Wilson loops. So I think most people know what Wilson loops are. integral of the connection around some loop, and I can also take a surface, and I can integrate the electric field, which is a one form, on the surface, which is, it's a, the electric field, sorry, is a vector density, which means it's dual to U minus one form, so I can integrate it on the surface, and the Wilson loops and the electric field integrated against surfaces simple Poisson algebra in the Poisson structure of the gauge theory, and that is called the loop surface algebra. Now the idea of this quantization is to quantize the gauge theory using that algebra, the fundamental algebra, rather than the canonical algebra of the A with the E, which is what's usually done in quantum field theory. And the reason, at the reason for doing it this way is this theorem, which is that there is a unique cyclic representation of the algebra of Wilson loops and electric fields smeared over surfaces whose Hilbert space carries a unitary representation of the diphenomorphism group. And that's called the Astrakar-Levandowski representation. And it's a rigorous theorem, and I'm not a rigorous mathematician, so I'm not going to explain the proof to you. But that's in contrast with Poincarean varying quantum field theories, where you start with the A and the E and its Poisson brackets, where there is, by the way, no such uniqueness theorem, because of various infrared issues. Now, this means that there is a unique diffeomorphism varying quantum field theory based on this

22:30 gauge group and the quantization of this algebra for each gauge group. And it's that uniqueness on which the theory depends. Now, what one then does to get to the theory that is defined is using that unitary representation of the diphenomorphism group, one then can go down to a space of diphenomorphism in varying states. So here, the states are labeled by loops in the manifold, created by these Wilson loop operators, one then goes down to the space of diphenomorphism in varying states, which live in the dual of that Hilbert space, which are labeled by diphthymorphism classes of loops. And again, I could give a course on how all this works together, but I just want to give you the basic orientation and go on and talk about recent results. Now, what does this have to do with gravity? The main discovery of Amitabha Santana by Ashdakar is, and Tobansky before them, but nobody understood Tobansky, general relativity can be understood as a diphthenomorphism in varying gauge theory in which the self-dual part in 3 plus 1 dimensions in which the self-dual part of the space-time connection is the configuration variable, and the frame field is the conjugate electric field. That turns out to generalize, and in fact, every general activity in every dimension and supergravity in every dimension from 11 down to 3 to 2 plus 1 is understood as a gauge theory that is where the configuration variable is a connection rather than the metric. So all those theories can be quantized using this method, and the quantization at the kinematical level is unique because of this theorem. Thank you. There's a subtlety which I'm going to rely on a great deal in a few minutes. It turns out that if you turn on the cosmological constant, you're not actually precisely in the Hilbert space I just defined.

25:00 There's an infrared effect coming from turning on the cosmological constant. And the effect of turning on the cosmological constant is to take the Lie algebra. So in the case of 3 plus 1 quantum gravity or classical general relativity or supergravity, The gauge group of interest is SU2, but what I'm going to say is more general, but I'll just leave that case. Turning on a cosmological constant, quantum performs the algebra of SU2 to a Q given by this formula, where the level K, this is a standard formula, where the level K is inversely related to the cosmological constant in Paul units. and representing quadro group labels on these spin networks is a problem that's well understood and topological field theorists have understood it completely and by the way, these theories are closely related to topological field theories and there's a certain sense in which gravitational theories in every dimension can be understood as restrictions or perturbations of topological field theories A lot of the rigorous side of the subject comes through the study of topological field theories and one of the things that's well understood in the study of topological field theories is that to get all the phases right to represent the deformation of the gauge algebra to a quantum algebra, one has to label not graphs with representations, but ribbon graphs with representations. And this is because there are phases in the representation theory of the quantum group that have to do with twisting of the ribbons. And people who know conformal field theory also know this in a slightly different language. If you know conformal field theory, you'd be more likely to think of an edge as represented by a cylinder and a frame graph as a bunch of cylinders tied together, and then the labels given by the quantum group representations are the same thing as the conformal blocks on the related alpine algebra. But it's the same structure.

27:30 This looks strange because SU2, you expect this group because it's present at the classical level, but even in general activity, with Lananda, you have SU2 acting, not deformed SU2. Let me give you a physical argument, which I don't have a transparency about, but if you buy the holographic bound, then you believe that if there's a positive cosmological constant, there's a horizon, and the states that an observer can see on a horizon should have a finite dimensionality given by the area of the horizon over 4 in font units, and that turns out to be 3 pi over Newton's constant times 4. the cosmological constant is the number of states that you could expect to live on in cosmological horizons in De Sitter space. If you rotate, you should expect those states to rotate under an irreducible representation of the rotation group. But now you have a contradiction if you think you have a constant rotation group because you can have an arbitrarily large irreducible representation of the rotation group. If you require that you only have as large a representation of your rotation group, as you can represent states on your horizon desider space, you come to exactly this quantum deformation of SU2. That's not how it was derived. From our point of view, this is how the restriction on the states on the horizon desider space is derived. Excuse me, is that the same deformation than in double spatial relativity? No. But related to it, as I'll describe, let me not say no, possibly for reasons I'll describe if I get there, which is unlikely unless you give me a lot of time, but I can tell you that privately. if there's justice in the world so just some very quick results and then the new results sorry I know I contradicted myself three times I'll try not to do that again there exists in these theories semi-classical states these are old results excitations of the semi-classical states include long range These are theories that low energy approximation

30:00 have gravitons in the masses in two modes. Sums are related, so they're particular spin-phone models which are derived by, through relationship with pathological field theory and are considered rigorous colonization of general relativity in three plus one or higher dimensions. These are called Barrett-Crane models. And the sums, so in each history, can be seen as something like the Feynman diagram, where there are local moves analogies to vertices, and there are sums over the labels, which are not tied down by the labels of the initial and final states, analogies to the momentum integrals in Feynman diagram, and these are known in 3 plus 1, and that is in the Lorentzian signature and in the Euclidean signature, 4 dimensional, to be convergent, and there are results analytically and numerically that show that. Recently, Rodellian collaborators in 3 plus 1, well, 4 Euclidean dimensions, have derived the graviton propagator and checked that this law is true at long distances. 2 plus 1 gravity with matter is solved, given an effective theory on Kaplan-Makowski's space, which implies that double special relativity, that double special relativity or capital-clunk gray algebra used the correct low-energy symmetry, and that's work with Tredow and Nadine. There are many reduced models of carmology and black hole interiors which have recently been solved the last five years or so, showing us that space-by-singularities, at least in this class of models, are illuminated and placed by balances. that is even being extended to predictions of corrections to the CMB, which are a few percent, unfortunately a few percent, at one wavelength where there's, unfortunately, already a large uncertainty. I'm forgetting the name of the issue. There's the fact that we only see the universe once problem that Stephan Hoffman and Oliver Vinclair. And there's a lot of results about black philanthropy, including very recently results of Mohamed Ansari, which are about to appear showing that there are observable corrections to Hawking's spectrum, which was quite a surprise. All of these things are things I can say more about later.

32:30 But there's always been a criticism of this line of work. If you have so much right, if you have the unification of climate theory and general relativity or space-time right, then what about the rest of physics? And that's what I'm going to be talking about next. And this was a surprising work with Sundance Wilson-Thompson from Adelaide and Fortuny Martin-Guro. The hard questions in luponic gravity and spin-fall models have always been about How do you get to observable physics? What's the long-distance approximation to this dynamics of quantum geometry? Do you get general relativity back? Do you get quantum field theory back? Etc. Related to that, there's always been a puzzle which we ignore. And this is an embarrassment for people like myself and others who have been working on this subject for a long time. I told you that every embedding of every graph gives a distinct state. But in three dimensions, which is the world we may live in, there is braiding and knotting. So I don't know if everybody can see this, but supposing that I have, say, two nodes which share three edges. They could just be connected like that. Or they could be braided. say it like that. And there's an infinite number of ways that could be braided, giving rise to an infinite number of states. Now these states turn out to, from the point of view of measures of geometry, and I skipped this, but we know how to measure areas of surfaces and volumes of regions, and from that, from the point of view of those observables, These states are all degenerate. They don't change the quantum geometry. So what do they mean?

35:00 I'm not going to tell you the answer in a while, but for 15 years we didn't know the answer. More than 15 years. What's an observable in quantum gravity? Well, because of the issue of geomorphism invariance, an observable is locally a constant of motion. observable time differences at boundaries. So, there's a close relationship, at least locally, between observables and conserved quantities, both in classical general relativity and quantum geometry. So, what are the conserved quantities of these dynamical quantum geometries? How do we get to the long distance or low energy limit? In order to get to the physics we know, we have to talk about local. But, this physics has no background metric. non-local. How do we define excitations without a background? How do we define states that correspond to gravitons or other local excitations? Again, we know how to do it at the semi-classical level, but that's because at the semi-classical level we build states like coherent states around classical space-time geometries. How can we do it in the full Hilbert states? And And if we have expectations, how do we keep the quantum coherence? And this is an argument that Plotini's, and for me it's a clear physical argument, so let me start here to introduce their work. How is it that if we make a photon, as one can see in laboratories, It can travel coherently for meters or for megaparsecs with its quantum phase information coherence in quantum field theory. In quantum field theory, we know the answer. It can because there's a symmetry that protects the excitation from decohering by interacting with the vacuum. The vacuum is called full of quantum fluctuation, but the vacuum is in the singlet of the Poincaré group, is in another representation, and therefore the photon stays coherent. But what about in quantum geometry? If you believe that the world at small scale is this, you know,

37:30 is this zero position of quantum geometries, which are ever-evolving, then at a small scale there is no Poincare invariance. Poincare invariance is only, at best, the symmetry of the semi-classical limit of the ground state. So how is it that an excitation propagates through quantum geometry without decoherence into the noise of quantum geometry? And that's the question between our collaborators ask, and they propose an answer. And the answer that she proposes is that we can understand these questions analogous to the questions that people who want to make quantum computers ask. And maybe this is because of Perimeter we have also a lot of people thinking about how to make a quantum computer. But similarly, the problem for a quantum computer is how to create an excitation in some solid or whatever that propagates coherently even in spite of the fact that it's in interaction with a lot of stuff. And the interesting thing is that they have language and methods investigating these problems which go deep into the Hilbert space, which are not semi-classical. We also, in quantum gravity, are deep in the Hilbert space and not semi-classical. So one of the methods is called the methods of noiseless subsistence, or other people call them decoherence-free subsistence. And the idea is that there are emergent symmetries which can protect excitation of subsystems from decohering through noise between those subsystems and the larger system. And there are many examples, and again, and I'm not an expert, so I'm not going to give the class of examples from quantum information theory, but the idea roughly is that you can identify a subsystem in a big Hilbert space in such a way that you can identify a symmetry that the noise acts with respect to, and then you can study irreducible representations of those symmetries and isolate little sectors where there's climate information that the noise is invisible to, because it's protected by a symmetry.

40:00 And those are called decoherent-free subsystems. And the strategy of the Plotinian Collaborative is to identify sub-sub-systems and then to study the symmetries to protect them in different states and to identify the quantum state associated with Minkowski space-time or Desider space-time as the state that has symmetries protecting the coherent excitation which have Desider or Poincaré invariance or some deformation of them. So that's the research program. And what I'm going to talk about is an offshoot of it, which just comes from really the first step, which is identifying emergent degrees of freedom. So, there are two results. One is their result, and then the result from our collaboration also with Sundance, Wilkins, Thompson. is that a large class of the theory, a large subclass of the class of theories I define, have noiseless subsystems that can be interpreted as local excitations. And the result that we have with Sundance and Focini is that there is even a class of such theories where the simplest such coherent excitations match the fermions of the snare and motto. And you should be going no way, you should be skeptical at this point, We'll show you. I'll show you why we're able to say that. I don't know what you call a subsystem. Is that the set of states of Hilbert's space? Yes. It's a... you make... one finds a tensor product... Since I'm not an expert on that, I will try to say it right, but then I'll show you an example. one tries to find a tensor product decomposition of the Big-Hilbert space such that one factor in the tensor product would describe a subsystem, such that a subsystem of that evolves decoherently with respect to noise from the outside. Okay, so we're going to study a class of theories defined by frame graphs, So these are called by the mathematicians ribbon graphs. For simplicity, I'll study here just trivalent, so they're tied together on circles, or disks, like this.

42:30 So there's a basis state for every oriented twisted ribbon graph embedded for precision in S3 of the topology class. We're not going to worry. For these, topological equivalents are the same as diphtymorphic equivalents. As far as the choice of the quantum group that labels these, it turns out not to matter for anything that I'm going to tell you. It actually turns out not to matter at all, but we assume there is some. The dynamics, so that's the Hilbert space, the space, the dynamics is going to be given by the two fundamental moves on trivalent ribbon graphs, which are exchange moves, and expansion moves. This is the ribbon version of a node close to a triangle. One way to see these graphs familiar from planets from conformal field theory language is that there are trinians. So each one of those, that's the little trinian there. It has three edges where it gets tied to other trinians, and where you put representation labels on the edges. And a written graph is a bunch of trillions then tied together for people who know that language, and then labeled by some conformal glossary. And it's going to turn out not to matter what the evolution amplitudes are so long that there are amplitudes for these loops. So this is a very generic class of theories, and the talk is titled Generic Results. So, if this is not a result that is required with the Baratrain model or anything like that, okay? Questions, are there invariants under these moves of the states I've described, and what are the simplest states preserved by those moves? So, we go, in turn, the first set of moves, the answer... So, here are the two sets of moves. Here's just an example of what might start with an embedding of a rhythm graph like this. This is what Lubrish Motol calls the octopus state. And by the local moves it can evolve, and one of the themes, one of the points is that,

45:00 the, well, you'll see, but it can evolve to become just, you know, quite complicated in a small number of steps. So, are there invariants that are preserved under those local moves? And actually, it's not too hard to show that there are. So, let's just look at the simplest exchange moves. And there's an obvious observation, which is that the topology of the embedding of the ribbon graph remains unchanged here. So any invariant of the topology of the embedding of the ribbon graph is a concept of motion just for this move. And here's one that we find useful. There are many, but this is sufficient for the job What we're going to do is to consider the edges of the ribbons as curves embedded in S3 and consider just the topological class of the embeddings of the curves in S3, that is the link class of the edges in S3. That's called finite position to link to the ribbon. But we want invariance also under the rule where one takes a node to three nodes tied together like a triangle, and what that does for the link, that is for the edges, is it creates a circle, and that circle, or it eliminates a circle going this way, either creates a circle or eliminates a circle, and that circle is always unknotted and unlinked with anything else. there's an invariant which is just removed while the unknotted, unlinked circles. And that's called the reduced length. And that, I've just shown you, is an invariant of these states under local rules. That is, there's an operator in the Hilbert space that I've defined which measures that. Now, even better, this allows us a definition of a subsystem, which is a key part of the discussion. So consider, for example, this embedded ribbon, and you can see that blue curve that identifies an inner region, which is a subsystem, and what we mean by a subsystem is that there's a factor in the edges, in the link, which is disconnected from everything else.

47:30 So this factor here, in the link of the embedding, is unlinked and unlinked to anything else and that is going to define a subsystem. That is, its topological properties define operators which just measure what's happening in that subsystem. So for example, here is the reduced link of that embedding, and it's easy to show that it is preserved under evolution. So we have operators that, and I'm not showing you the details just why the results are true, we have operators that measure conserved quantities associated to subsystems of these embeddings and these graphs. Another invariant is chirality, and it's interesting to notice that, and occasionally people have wondered about this, that knots and links and rays have chirality, and so under a parity transformation, So the manifold has a differential structure. Let's just consider oriented manifolds, then there's a parity transformation. And under that parity transformation, typically braids go to braids that they're not dimorphous to. And so the invariants we have are both the chirality, which manages the properties under chirality, and the reduced length, and those together are sufficient to distinguish twists, so an untwisted edge from a twisted left edge and a twisted right edge, so we can distinguish left and right-handed structures. So the invariants I just defined, let me just emphasize, of the labeling algebra, the quantum root G, and the choice of the evolution aptitudes. But for every theory defined by a G in evolution aptitudes, there exist these invariants. Okay, so now we said to ourselves, this is not the way it really happened, of course,

50:00 but I'm going to make it up, to make it head into physical. We said to ourselves, what are the simplest subsistence with non-trivial such invariants? And we found a simple set which had to do with braids on n strands to find the following way. So we have here a graph, we have a bunch of nodes, that is a bunch of trinians glued together, with edges coming out of them, another one here. At least one, and possibly both, are connected to the rest of the graph. There's some big graph outside the system, and here we have a braid. to anything that is just a braid. So that is, these edges here can be twisted and they can be braided. Now, given AND, which defines the number of strands in the ribbons here, there is a group called the braid group, which can take any braid, starting with the totally unbraided, untwisted one, can take it to any other one. That's generated by a series of moves. So here's D1, which crosses the first two, D1 inverse, D2 which crosses the second two, D2 inverse, etc. And there are moves which twist to the left and the right. So to each braid in here, there's a group element which is a product of braiding and twisting, which produces it. And therefore, there's a notion of charge conjugation, which is, if there's a state here which is gotten from a trivial state by a group element, there's another state which is gotten by the inverse group element, which we call the charge conjugated state. So this is a kind of background-independent notion of charge conjugation. Okay, so now we want to classify the simplest subspace, and we have a charge conjugation quantum number, a parity quantum number, and some other quantum numbers. So what do we have? And it turns out that the simplest non-trivial subsystem is a three-strand braid, and the simplest ones are the ones with two crossings. This is just to demonstrate that the reduced link is non-trivial, so there is indeed a structure there,

52:30 which is propagated and conserved under the local rules. And this comes in a left-handed version and a right-handed version. One then classifies the invariance associated with different such structures by computing the reduced link. differ just by twisting. So we have a left three-strand, two-crossing braid, and a right one, and we want to classify the different things that can happen under twisting the edges. That's all this left. And it turns out that this problem was already studied. And the problem was studied by Sundance Wilson-Thompson in a paper of a year ago, April, in H-E-P-P-H, which I never read, but for some reason one day I looked at H-E-P-P-H. and he showed in this very remarkable paper, I think very remarkable paper, that the classification problem that I just find is coming from quantum gravity is equivalent to the problem of counting states in a prion model and because there are some younger people here, I found that I have to, even with very good people in particle theory and string theory and so forth. I have to define what a prion model is, because they haven't been spoken about for 20 years or so, but most people. So, a prion model were, in the 1970s, models of substructure of quarks and leptons, in which it turned out that there are simple gains by which, given just three elementary entities, called preons, a neutral preon, a charge plus one-third preon, and a charge minus one-third preon, which is an antiparticle of the charge plus one-third preon. You can, by putting them together in triplets, according to certain rules, you can reproduce all the quarks and leptons of the standard model. and unfortunately no dynamics was ever found in the world of gauge theories or super-symmetric gauge theories

55:00 which allowed one to go further with this observation and the rules were rather peculiar and therefore they were forgotten but what Wilson Thompson showed is that the rule, one of the rules is naturally explained in terms of a relationship to a counting problem of grades, and one remains mysterious and remains mysterious for us as well. So I'm going to tell you the rules. So the idea is that a twist is associated with a charge in units of one-third. P and C in particle physics, we just identify with P and C. We identify this elementary entity as just a strand here. into a triplet we say is in one of these coherent states associated with three-strand braids. And for those of you who know about preon models, there was a mysterious thing in which one had to distinguish in a way that was not consistent with either fermion or boson statistics the position that you were in of the three. And that turns out to be naturally explained by physicians in the braid. An equivalent thing that, to my knowledge, nobody ever said is that the prion models only make sense if you give the prions anionic statistics, which is equivalent to saying that they're classified by physicians in the braid. That's something that maybe somebody said somewhere in the literature, but I haven't been able to find it. Now, there's a mysterious rule, which is that there's no triplet So we do not consider the states with both left- and right-handed twists, and this was mysterious in the PR models, and so far it's mysterious to us now, to us still, at some dynamical consequence, but I'll come to the things we don't know, which is a wrong list. Okay, so my next study is not a classification problem, and here we go, two crossing left-handed in varying grades. There's the one with no twist, and the one with no twist by the interpretation has no charge. And we go on. Then there's the one with three twists, one twist on each edge. So that has charge plus one or minus one. And we go on. Then there's the ones with one

57:30 And if there's one of the edges twisted, then it's a distinct state, and you have to do the computation, assure yourself that it's a distinct state, for each of the three ways of each of the three edges being twisted. And so therefore, the fractionally charged, Mark, this is the punchline, the fractionally charged states are the states that come in triplicate, which is the thing that the prion models explain. So you get a triplication, you get in triplicate the charge one-third states. So again, something that's always been, fine, always been mysterious in the standard model is a relationship between the one-third of the fractional charges and the three of colors is naturally explained in the Creon model and is naturally explained here. And the same thing is with the charge two-thirds. So it's natural to give those assignments. Now, there are also the right-handed ones. You consider also all the parity and charge conjugated duels of those. enough for the uncharged, unchoisted one, only for that one, charged conjugation is equal to parity conjugation, and therefore there's only two states associated with neutrino and not four. So one has to believe that this, which we're going to call the lowest generation, has still a massless neutrino, because there are only two neutrino states, and one gets the fifteen states of the first generation. Here's the right-handed version, where they all are. I'm going to speed up a little bit. So the emergent symmetries include, they're larger than this, but they include an SU2 left, an SU2 right, a U1, which counts the twisting, an SU3, which exchanges, parity charge conjugation when gets right, two neutrino states, four fully charged states, and fractionally charged states. This is not dependent on the fact that you have the S3 embedding.

1:00:00 No. But it's dependent on the fact that you have a three-dimensional embedding. Yes. So one other three is linked to the other two three. Now what about higher generations? So for higher generations, those we presume have to do with higher numbers of crossings. And interestingly enough, if the generation number counts the number of crossings, then for the higher generations, you have additional neutral states, for example this one, so you have a possibility of mixing and therefore explaining the mass differences among neutrinos with mixing with these states. This is in progress and not done, so we don't have any claims to make about this, but when it's naturally a role for mixing of neutrino masses in higher generations. There's a lot that we don't know, so any question that you might ask, we don't know the answer to. And maybe to go on, can I go on these types of typology things, quickly? Okay. I hope so. Okay. Let me at least give you the flavor of the cosmology thing. So I will talk for five minutes, he said I had ten minutes, about a few minutes ago, motivation, and then I'll present a very simple model. So one consequence of the approach to how physics emerges through this part of information idea is that locality of the space-time should emerge in terms of the interactions of these excitations. And if they do, then they can expect to be what we call disordered locality, the underlying graph may not match completely, locality in terms of the underlying graphs may not match locality in the emergent space, and we have a number of different arguments for this, and causality of the underlying microscopic theory may not match the causality in the emergent low-energy limit, and we call this disordered locality. There are a number of different arguments that lead us to this, and first of all, that the graph state presumably involves

1:02:30 two repositions of large numbers of states, each given by a different graph, so why should all the different notions of locality match? There's also, let me just show you this, supposing that you get them to match this, supposing you start the universe off in a state where as it grows up by these open rules you get a state dominated by one graph which looks something like a lattice or a random lattice where locality in the macroscopic coarse-grained sense matches locality in the microscopic make that work by a clever choice of the initial state. But that's a very unstable property because all I have to do is pick any two nodes in the micro-state and connect them. And I just made a small change microscopically, but I messed up locality macroscopically. So, exactly matching micro and macro locality is very unstable in these background-independent theories. And indeed, Bottini has been arguing this for a long time, and most of us have been resisting because we thought this was the definite theory, but nobody ever gave really a good answer. So we began to study the possibility that actually they're there. And what they're there means is that the state of the world, the graph that dominates the ground state or the graphs that dominates the ground state are similar to what statistical physicists call small world networks. That is, there's some graph which is local in definition, that is, nodes are only connected to other nodes that are order Planck's length away, but that locality is disordered by a number of random non-local connections. Well, it turns out that statistical mechanics on such graphs are recently the subject of a lot of studies in statistical physics that are called small-world networks. And the simplest hypothesis, if this blew up, and there's been some numerical study of this by some of our students,

1:05:00 if this blew up from a small clock scale quantum geometry, is that what happens is these non-local connections get frozen in. That is, the local moves only rarely affect them. And I'll tell you in a minute how they affect them. So in general, they're frozen in. and therefore there's no natural length scale that should characterize the distribution of these non-local connections so we can assume that they're scaling varying up to the Hubble scale now. Now it turns out that there's been study of statistical mechanics on these things, I can talk about this if people want, and they show that actually we can measure, if there are not too many of these, and I'll come at the end to how many is too many, non-modal connections by measuring average endpoint functions. Another way to put your mind in these is, you know, supposing that there's a billion Hanck-scale nodes in this room connected to random points in the universe, but there's a very small amplitude of order E-cube times Newton's constant for any photon to leak across one in this room. You can think about, if you're bored by what I'm going to say in the next few minutes, actually those non-local connections are there. So the proposal is that disorder in the real world is, excuse me, that causality is disordered, and there are actually quite a few, and we'll talk about in a minute, how many of these non-local connections between Poncio nodes in the present universe and far away nodes. So here's a model. A model of this is Friedman-Robertson-Walker, but we're going to disorder locality, so we're going to choose random pairs of points from the spatial manifold within the co-loving volume and identify them. Microscopically, that's because there's some microscopic state underlying the flat within this flat Friedman-Robertson-Walker and there are these non-local connections. So there's some probability if X and Y are points in space as a function of A in the scale factor, that there's a non-mobile connection between those points, between a unifying around X and a unifying around Y. And this is going to be the key quantity in the model.

1:07:30 as a function of the scale A, by scale invariance, this goes just by, this goes 1 over the volume squared, so 1 over A to the 6, times just the number of these non-level connections as a function of the scale factor, which is a number of non-level connections just given by the migrating device over the probability. Okay? V is the volume of sigma, right? It's flat, so it doesn't matter whether it's a periodic boundary condition or not. It turns out, the studies in statistical mechanics have shown us that there's a useful approximation, good for such three-dimensional models, which is the annealing approximation, which is that a random distribution of identified points of this kind, with a probability given, as I did, has the same effect on the energetics of the model, putting a small non-local coupling between every pair of points of strength beta, which is P over the volume squared. So we're going to do just the energetics, the thermodynamics of the system, and we're going to replace this rare distribution of non-local connections with just a tiny coupling between every pair of points. And there are calculations and arguments that assure us that this is a good of consummation. What this means, and I'm almost at the last transparency, believe it or not, is that one corrects the ordinary local freedom of Robinson-Walker dynamics by a non-local connection, and Sigma is then referred to a degree of freedom of interest, really any degree of freedom of the system, and we're going to highlight the non-local effect, what the non-local So the idea is the following. If I have for any degree of freedom, whether it's the quantum geometry degrees of freedom or matter degrees of freedom, something that you would recognize as the nearest neighbor coupling, then there must also be that nearest neighbor coupling across these identifying non-local points. So if there's a coupling like sigma dot sigma across nearest neighbors in some lattice representing flat space, then there's also such a nearest neighbor coupling across every non-local length.

1:10:00 And I'm going to study here anti-paramagnetic couplings. The reason why I'm shooting anti-paramagnetic is work of Ansari and Marco Kula where they show that the evolution moves in the quantum gravity models can be understood in a certain approximation as anti-ferromagnetic couplants. Because if you make an expansion here, you want to make a contraction there, and vice versa. But I'm just really, you can really think that there's some other field which has an anti-ferromagnetic couplants. We'll come back at the end if I have time, which will be in two minutes, to see what happens if that's a magnetic couplants. That gives rise to an approximate, no, to a continual picture where we integrate twice over space and time with some probability distribution of connections, and the sigma, the local sigma fields, are given, are coupled. So this is what the effect looks like macroscopically, and P is the probability distribution, and the simplest assumption here is that the Ransom variance is just broken by these non-local couplings, so there is some preferred time in which these non-mobile cufflinks are coincident. One can investigate Lorentz-invarying versions in which the non-mobile cufflinks are more Lorentz-invarying, but this is the simplest model. One has to study how these non-mobile connections evolve. There turn out to be rare processes by which a non-mobile connection gives rise to two, Another rare process is whereby they annihilate each other. We assume that these are in equilibrium. So I'm going quickly, there are the processes by which they breathe, the inverse of the processes by which they come into us, by which they disappear. And assuming that they are in equilibrium tells us that the number of roles like the volume squared. That was very quick, down to the end. That's a simple statistical counter. and one puts that in and here's the basic physics the basic physics is that under the approximation where every degree of freedom has a tiny coupling to every other of its degrees of freedom everywhere else in space that mean field theory is exact every degree of freedom is coupled to the average

1:12:30 of all the others in space and that means that there is a contribution the effective energy, which goes like, let me come down to it here, it comes down to the mean field squared. And that's the calculation, but that's the outcome, is that there's an effective contribution to the energy that comes down to the mean field squared, with a certain mass squared, and that is, we propose the vacuum energy. That is, a symptom of these non-mobile connections is for every fundamental degree of freedom there is a mean field of contribution which comes down to the average value squared. And one can work out, in terms of the non-mobile connections, what the mass square is. And the interesting thing is that if it's a gravitational degree of freedom, and I'll just say this and stop, then the number of non-mobile connections you need in the horizon scale, present horizon scale is 10 to the 60, to give the density of non-mobile connections for 10 to the minus 120. If it's a matter degree of freedom, there's another factor. means constant, and you need many more, you need one per permit. So, I think I've run over. Let me summarize and say what I did say, and then close. So, summary, I've been talking about a class, a large class of quantum gravity theories. They were motivated by the chronization of general relativity and supergravity and related theories, but once you state them, they're very generic. And since quantum theory is fundamental, let's just study them. They're given by dynamics of embeddings of labelled graphs evolving under local rules. There's lots of indications that these are quantum periods of gravities, that they have semiconscript states, that they have gravitons, But what I talked about today is not that. What I talked about today is the surprises coming out of the work. And I talked about two generic surprises. One of them is that these models, all or many, have elementary particles in them,

1:15:00 that they have expectations that can be identified as coherent, noise-free subsystems that propagate coherently. And I showed you one adventurous result, that counting the simplest of these in a certain class of models brings us on top of prion models, which are plausible models of substructure of elementary particles. And what was hard to explain in the prion models, which is the dynamics that bound the prions together, is now explained in these models that bound the prions together are these topological conservation laws coming from quantum geometry. So it's a new twist, if you will, on prion models. Then I talked about another surprising consequence of these models, which is that if microscopically locality arises from these discrete structures and causality as well, then it's very natural for there to be dislocations or disorder in locality, in the low energy classical limit. This could, if they were too prevalent, kill the theory. But at low levels, it turns out to be helpful, possibly, and gives a natural explanation for the vacuum energy. What I didn't have time to say is it gives, of course, a natural solution to the horizon problem, and gives, because the distribution is scale-of-variant, it turns out, a natural approach to getting a scale-of-variant distribution of fluctuations in cosmology. So, quantum gravity is an adventurous field. Its main problem has been and continues to be a lack of contact with experiment, but what I think that these new results are doing is they're allowing us at this stage to begin to make connections with real physics and perhaps, if we're fortunate, make genuine predictions for experiments. So thank you.

1:17:30 I have one question. I'm not sure I followed the last point. I went very fast. I apologize for the response to me. Would that be true to say that this is due to an effect of quantum correlation which remains at large scale for some degrees of freedom which has something to do with gravity? This is that. So this would be completely semi-classical. Well, the interesting thing is that it's very low energy. and long distance, but at the same time, it's quantum mechanical. I mean, here's what you're speaking, here's one picture that we use. The world starts, supposing that the world is described by some discrete quantum theory like this. The world starts out with some discrete structure abroad. It starts out, whether it starts out from a bounce or who knows why, it starts out at a scale below classical scales. So there's no reason why the initial state should know about any principle about being classical at large distances. It should become classical as it blows up. So we start with a random graph, and Hal Finkel, for example, one of our students, has studied this in some detail. random graph, say 100 nodes, and then grow it up by local moves. As you grow it up by local moves, that is by point to triangle, point to tetrahedra, you begin to get local order over some regions. And those regions where you have local order grows. If making triangles, if it's exchanged, if it moves that makes triangles dominate, then you get exponential growth, something like inflation, of these regions that have local structure. But you don't kill the original random connections. So in these regions which are growing local structure, they're still connected by initial random connections. And since the dynamics is only by local moves,

1:20:00 they get frozen in. and so what one gets naturally is what we were hoping and what I think everybody in the subject has been implicitly assuming is that if you grow by local moves you'll get a structure which is mostly local and if there are deviations from locality that is, if it's impossible to put some coarse-grained metric over the whole thing and describe it by some classical metric those deviations from macro-locality will be plot-scale you could sort of average over things and you would get deviation of constant. But what happens in the simulations, which are, of course, classical simulations, but nonetheless what happens is that these non-mobile connections get frozen in and stay there. They do occasionally evolve, and let me just, I went through this very quickly, but, for example, this is some non-mobile connection connecting two regions here and here arbitrarily far away, non-local means the return, the smallest closed loop, this link in the graph is in, is very large. Then there is an exchange move, there is a move that's local in the graph that acts over this non-local connection and produces two non-local connections. So that's rare that the move acts exactly there, but it will happen sometime and so it increases the number of non-local edges. And the result seems to be that, as I argued, that the system reaches a kind of equilibrium where the inverse move is balanced by the move that creates these non-local connections. The inverse move requires that two non-local edges basically line up and then they can be annihilated by something that happens at one end and bringing it back to that. And they're always there at some small level. Now exactly the small level depends on the micro dynamics, it depends on the different amplitudes for the different kinds of transitions. So we don't try to estimate that, but we notice that in equilibrium the number of such number of dimension scales like the Bind script, so like any of the sticks. So that's the thinking that we have about these things.

1:22:30 This is tentative, on the other hand, and there have been sort of, what we can say sort of goes to both sides. Nobody has been able to, in spite of the fact that this has been talked about for some years, it doesn't seem that there's any dynamical assumption or assumption about the initial state that makes locality macroscopically coincide with locality macroscopically. So they have to be taken seriously. Clearly, if there's too many of them, then the world never goes to a local state. There's never a notion of locality. And we don't know why that's a question that we don't have a satisfaction answer. So one of the problems is one of you never gets invariants with both left and right twists. So the question is this, if you look at the braid group, so the braid group has V1, V2, and the t operator, but this thing, yeah, you have somewhere in the slide, so if you choose a basis of local moves, you know, you choose some system of local moves in the graph, these local moves are going to tell you how you realize the b1, b2, and the t operator, so the question is to have an idea if you can, whether you can choose a good basis of local local moves come with amplitudes. Maybe you can choose a basis of local moves where some amplitude will be sort of suppressed or something that will effectively show you why you don't get those twists at the same time. Well, right. And there's a number of questions which are related to that. So let me just confess up to what the next hard problem is. The local moves that I described lead to the crime numbers that I described being absolutely conservative. So the question is, do these

1:25:00 states which have been so identified can they then interact? Okay, question one. Question two, are the interactions going to be naturally describable in some low-energy approximation as gauge interactions? That is, are these symmetries, if I have one subsystem here with these symmetries and another one, are the symmetries local? So that the interaction must, in the end, be describable in terms of the gauge theory in some effect of low energy's limits. Related question, assume that we're in a state, and this hasn't been shown, but assume that we're in a state where these propagate coherently in the larger limit through space. So these things are expectations sitting in a state which otherwise has translation invariance to some approximation, and therefore could be regarded as also having conserved momenta. Can we compute the math matrix, which is closely related to the question of is the chirality preserved or can left-handed states become right-handed states, in which case the chirality is not conserved, in which case everybody becomes massive at the Planck scale. So we here have the same problem as everybody else, which is we would like, either through a Higgs, some composite Higgs or some other way, that these states get masses, but we would not like them to get masses at the natural scale of it here. It's the same problem that string theories have and, you know, everybody has, that we unify with gravity and the national mass scales upon the mass. So, the mass matrices should be governed by, you know, large ratios such that the mixing of chiral states is very small on the fundamental scale of the model. and that's a big challenge that's probably the big challenge because if all the states get the font energy then it's pretty but it's useless so we have ideas about all these things and I'm happy to entertain people or you know

1:27:30 hear people's ideas I'm happy to confess our ideas you know over beer or wine or something like that, but we don't have results so far about these questions. These are the big questions, and there are many ways to say it, you know, how and why is chirosymmetry grade, etc. And there are some interesting ideas, there are observations that various people have made, but so far there's no results. And if there are no further results along that line, then this certainly is not going to turn out to be a political direction. You mentioned that algebra labeling of spin network does matter, and I wondered how much of property manifold you fixed does matter. I guess, if you fix a manifold, it constrains how you have an accessible algebra. I don't know that it does. So the topology of the manifold, I mean, presumably does not matter, but you don't know that. And differential structure might not really, or you just... The three dimensions is unique, so it can't. In the questions that I've just been asking about breaking a parallel invariance, where the mass majors might come from and so forth, the details may matter, and that's good. all of the true methods, I guess, you know, for this, I don't say, collection, distribution. Yeah, but I mean, we try to take the simplest assumptions. So I don't think so. I don't think it's going to require... Getting particle physics right in the miracle that actually comes out right is not going to depend on I'm choosing some strange manifold to three space. So I'd just rather work in R3 or S3. A very short question. I am not a specialist performing this film. But when you make the respond on a bit of fermion on your web. You will talk about the Nucleum, right?

1:30:00 And does it mean, does it suppose that the Nucleum is Majorana or Dirac or have nothing to do with? At this stage, but again, I clashed it too quickly, but in the list of things, so in my confessions, what is my confession taken? No, no, before. We don't know. We don't know, we don't know precisely that they are, we know that they're chiral states. So if they are fermions, they're vile. There is a small out because there are chiral vectors in three dimensions, E plus IB. And this we're also working on, and this we're closer. So I do believe that they're vile fermions, but we don't have a complete demonstration yet, so I'm not showing it. Do you reply to my questions, Tom? Could you say a little more about the early universe inflation? What that looks like in this class of theory? Um... I am... I am... I am... I am... I am... I am... I am... I am... I am... I am... I am... That tears also... What the army was saying so hard. Somebody else. How these non-local connections are in this form? Yes, yes. Again, that's in progress. The only thing I can show that has any substance that I didn't show is the estimate of the spectrum of fluctuations. Let me find that. I'm actually the boss.

1:32:30 So I think that this is entirely trivial, and by the way, it's surprising to us that somebody from the brain world or something like that has not made a similar observation. So this is just an estimate of the spectrum of the two-point correlation function delta rho over rho correlated with delta rho over rho. The assumption is that if there is a non-mobile connection between two points, and there's a fluct... So we have the same setup, and now we have a thermal bath of photons on top of it at some temperature. So in some local region, there's some fluctuation in photon number, and the assumption is that if there's a fluctuation up or down in the photon number in a region encompassing the mouth of one of these, these things could be seen like wormholes, then there's a small amplitude that's correlated over here, and the estimate we use is just g t squared, or l font squared t squared for the fluctuation. That's, it's unlikely, we can't think of any reason it would be larger than that, it might be smaller than that. So then you get a contribution to the correlation function proportional to that, proportional to the spectrum of fluctuation, this is the standard thing, times a sum over...